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Can the elements on the main diagonal of U be zero? If not, why?

Diagonal Matrices with Zero on DiagonalDiagonally Dominant Matrix Preserved after Gaussian Elimination (with a modification)Why do eigenvalues exclusively form the main diagonal in a diagonalizable matrix?Computing $PAQ = LU$ using Gaussian elimination with complete pivotingInverse of a matrix with main diagonal elements approaching infinityDoes encountering a zero pivot during Gaussian elimination imply that the matrix is singular?Show that the diagonal elements are not all $0$On the main diagonal of a particular matrix $(A+A^-1)^-1$Show that partial pivoting leads to an $LU$ decomposition of $PA$.LU decomposition of SPD matrix without partial pivoting?

-1

$begingroup$

Suppose that Gaussian elimination can be done on this matrix without pivoting. It results in an LU decomposition of $A$, i.e. $A=LU$. Can the elements on the main diagonal of U be zero? If not, why?

linear-algebra

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asked Mar 29 at 2:20

dxdydzdxdydz

47610

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If the diagonal entries of $U$ are zero, the matrix $A = LU$ will be singular, and so the matrix equation $Ax = LUx = b$ is not guaranteed to be solvable.
  $endgroup$
  – D.B.
  Mar 29 at 2:29
  

  
  
  
  

add a comment | 

-1

$begingroup$

Suppose that Gaussian elimination can be done on this matrix without pivoting. It results in an LU decomposition of $A$, i.e. $A=LU$. Can the elements on the main diagonal of U be zero? If not, why?

linear-algebra

share|cite|improve this question

asked Mar 29 at 2:20

dxdydzdxdydz

47610

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If the diagonal entries of $U$ are zero, the matrix $A = LU$ will be singular, and so the matrix equation $Ax = LUx = b$ is not guaranteed to be solvable.
  $endgroup$
  – D.B.
  Mar 29 at 2:29
  

  
  
  
  

add a comment | 

$begingroup$

Suppose that Gaussian elimination can be done on this matrix without pivoting. It results in an LU decomposition of $A$, i.e. $A=LU$. Can the elements on the main diagonal of U be zero? If not, why?

linear-algebra

share|cite|improve this question

asked Mar 29 at 2:20

dxdydzdxdydz

47610

$endgroup$

share|cite|improve this question

asked Mar 29 at 2:20

dxdydzdxdydz

47610

share|cite|improve this question

asked Mar 29 at 2:20

dxdydzdxdydz

47610

asked Mar 29 at 2:20

dxdydzdxdydz

47610

asked Mar 29 at 2:20

dxdydzdxdydz

47610